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A192798 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2. See Comments.

%I A192798 #18 Feb 18 2025 12:49:07
%S A192798 1,0,1,2,3,10,17,42,87,188,411,876,1907,4100,8863,19134,41289,89174,
%T A192798 192459,415542,897049,1936576,4180809,9025544,19484825,42064320,
%U A192798 90809993,196043706,423225563,913674090,1972469945,4258235410,9192822255
%N A192798 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2. See Comments.
%C A192798 For discussions of polynomial reduction, see A192232 and A192744.
%H A192798 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,0,-3,1,1).
%F A192798 a(n) = a(n-1)+3*a(n-2)-3*a(n-4)+a(n-5)+a(n-6).
%F A192798 G.f.: -x*(x-1)*(x+1)*(x^2+x-1)/(x^6+x^5-3*x^4+3*x^2+x-1). [_Colin Barker_, Jul 27 2012]
%e A192798 The first five polynomials p(n,x) and their reductions:
%e A192798 F1(x)=1 -> 1
%e A192798 F2(x)=x -> x
%e A192798 F3(x)=x^2+1 -> x^2+1
%e A192798 F4(x)=x^3+2x -> x^2+2x+2
%e A192798 F5(x)=x^4+3x^2+1 -> 4x^2+2*x+3, so that
%e A192798 A192798=(1,0,1,2,3,...), A192799=(0,1,0,2,2,...), A192800=(0,0,1,1,4,...)
%t A192798 q = x^3; s = x^2 + 2; z = 40;
%t A192798 p[n_, x_] := Fibonacci[n, x];
%t A192798 Table[Expand[p[n, x]], {n, 1, 7}]
%t A192798 reduce[{p1_, q_, s_, x_}] :=
%t A192798 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192798        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192798 t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
%t A192798 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192798 *)
%t A192798 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192798   (* A192799 *)
%t A192798 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
%t A192798   (* A192800 *)
%Y A192798 Cf. A192744, A192232, A192616, A192799, A192800.
%K A192798 nonn,easy
%O A192798 1,4
%A A192798 _Clark Kimberling_, Jul 10 2011
%E A192798 Comment in Mathematica code corrected by _Colin Barker_, Jul 27 2012